@nobody: "<quote>
but, this is REALLY REALLY interesting, and you can quite a lot from 
fractals - much more than you can learn from politics.  A few decades ago I 
wrote a mandelbrot set drawing program (see the "software I wrote" link on 
my web page, <url> if you want to DL it and play with it - it has 
several interesting options beyond the basic 2-D mandelbrot set.
Now, for you CNC guys - figure out how to machine something like this - 
wonderful stuff"
    @kg1: "<quote>
    Oh!  Now there's an idea for a fun project.  Take a tiny center 
drill or ball endmill, and some soft aluminum or machinable wax, and 
drip feed points from a PC program, maybe VB, that does the same thing 
pixel generating programs do.  Calculate points that are in the set, and 
put a spot in the work at every valid pair of XY coordinates.  Maybe, 
instead of using colors to represent how quickly a calculated point 
resolves itself, you could use Z depth, and get a 2 1/2 D relief of the 
2D graphic.
    Anybody got some spare time on a small 3-axis machine?
    I wonder what would happen on a 4-axis machine.  Calculate XY pixel 
locations, as above, but machine them as XC or YC on a rotary, with some 
scale factor so degrees would make sense in place of linear coordinates. 
  You'd get (I think) the normal shape wrapped around a cylindrical surface.
    What about 5 axes?  Plot AC instead of XY?  I can't even guess what 
that would look like.
KG"
    @nobody: ""Kirk Gordon" <email_address> wrote in message 
news:<email_address>
<quote>
if you want to try it, you are more than welcome to DL the program from my 
web site - the actual source code is just a few lines out of something over 
10,000 lines of turbo pascal - all the rest was to make a GUI, read the 
mouse, and so on - if the source code isn't there, drop me a note and I'll 
see if I can find a disk with it on it still - otherwise, it's pretty simple 
code - just square a complex number for up to a fixed amount of times - if 
the absolute value ever exceeds 4, stop.  select color based on number of 
iterations - at the time (mid 80s) I used 10,000 iterations - as you reduce 
iterations your figure gets smoother. "
    @Clhuprichguesswhat: "<quote>
  I doubt you used 10,000 iterations (though you may have
considered than many starting points) and always wonder
about truncation & rounding errors.
  With each iteration the number of significant digits needed
for an accurate calculation process (for THIS problem) just about 
doubles. 
  So you need extreme-precision math ... over about 4 iterations.
Cliff"
    @nobody: "<quote>
in this case, I'm sorry to say, I wrote the code.  It just squares (in 
double precision) a number up to the required number of times.  If you want 
to play with it, download it from my site - but honestly, you can't tell me 
how the code that I wrote and you have never seen works. "
    @Clhuprichguesswhat: "<quote>
  But I can tell you about the computations related to the 
Mandelbrot Set.
  Unless you can compute with all the significant digits you
don't know what is the "set" (to the level of your computation)
& what an artifact of your truncation & rounding errors.
  Arbitrary-precision arithmetic:
<url>
  Applications:
[
Arbitrary precision arithmetic is also used to compute fundamental mathematical
constants such as p to millions or more digits and to analyze the properties of
the digit strings (e.g. [1]), or more generally to investigate the precise
behaviour of functions such as the Riemann Zeta function where certain questions
are difficult to explore via analytical methods. Another example is in rendering
Fractal images with an extremely high magnification, such as those found in the
Mandelbrot set.
]
  Naturally, for coordinates for which the value went out of
bounds before that computation limit was reached you need not
worry.
<url>
Cliff"
    @Clhuprichguesswhat: "<quote>
  <url>
  "The History of Cabbage"
  "Although they appear very different, kale, cabbage, kohlrabi, cauliflower,
broccoli and Brussels sprouts are all the same species of plant."
  Lots of others from all over the world too.
Cliff"
    @someone: "news:hdvplm$a94$<email_address>
<quote>
want
me
No. Cliff knows everything about everything. How DARE you attempt to
discredit his knowledge about code that you wrote."
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